First, by way of summing up, I asked Reuben Hersh to please
address (what I called) the aboutness issue. This is his response:
> OK, how's this.
>> There is a definite, intersubjective, human, mathematical concept
> known as the right triangle.
>> It isn't about anything, it's an entity per se, not a name.
>> The Pythagorean theorem is about something. Namely, the
> aforesaid right triangle.
>> Mathematical objects aren't about anytghing, they just are
> intersubjective human concepts.
>> Mathematical statements are about something. Namely,
> mathematical objects.
Thank you. I am going to restate your above comments somewhat. I
hope I represent you correctly. You believe the following: (1) there are
"human, mathematical concept[s]" (the right triangle is one); (2) theorems
are about concepts (the Pythagorean theorem is about [the concept of]
right triangles); (3) there are no mathematical objects apart from
"intersubjective human concepts" (e.g., there is no "real" right triangle
somewhere, say in Plato's Heaven); (4) mathematical statements are about
mathematical objects (which are really intersubjective human concepts).
I hope the above is fair. I'm worried a bit about (3) (and about
some others to a lesser extent). You say above:
> Mathematical objects aren't about anything, they just are
> intersubjective human concepts.
which I've recast as:
(3) there are no mathematical
objects apart from "intersubjective human concepts" (e.g., there is
no "real" right triangle somewhere, say in Plato's Heaven)
I hope this is correct.
Anyway, I don't have any dispute with these four theses. They do
not seem too distant from some views others on this list have presented,
and I thank you again for explaining yourself. I would suppose, given
these four principles, that the real novelty in your view pertains to the
weight you place on human intersubjectivity and consensus. Your view
emphasizes the "agreement aspect" of mathematics (rather than the
"aboutness aspect"), which I assume you feel has been to some extent
previously neglected. Also, if it is your view that math is about our
intersubjective (human) concepts, then I am supposing you believe that a
way to better understand these concepts would be to look more closely at
the nature of social consensus among mathematicians.
Thank you for your comments. Please tell me if the above does you
any injustice.
Charlie
Smith College